I've got two small unlabeled plastic syringes with white cooling paste
- these are probably 8 years old. Should I throw them out and buy new?
I would, since a new tube of medium-quality thermal paste costs $5... And not for anything, but I wouldn't put 8 year old thermal paste anywhere near a new computer.
I'll need some paste to build a mid-range system, so I don't want
(need?) the best and most expensive cooling paste. If unused cooling
paste doesn't deteriorate over time, then I could even use what I've
Yes, you could use what you have. It probably still works fine, and as long as it's not dried out and was stored properly (i.e. at room temperature in the dark). It should perform just as well.
That being said, you should highly consider buying new thermal paste, or at least a reasonably priced, decent quality one. I never recommend anyone cheaps out on thermal paste, regardless of how much your system costs. For $5-10, you get a lot better returns on cooling performance, which translates into longevity of your system's components.